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To accelerate the existing Broad Learning System (BLS) for new added nodes in[7], we extend the inverse Cholesky factorization in [10] to deduce anefficient inverse Cholesky factorization for a Hermitian matrix partitionedinto 2 * 2 blocks, which is utilized to develop the proposed BLS algorithm 1.The proposed BLS algorithm 1 compute the ridge solution (i.e, the outputweights) from the inverse Cholesky factor of the Hermitian matrix in the ridgeinverse, and update the inverse Cholesky factor efficiently. From the proposedBLS algorithm 1, we deduce the proposed ridge inverse, which can be obtainedfrom the generalized inverse in [7] by just change one matrix in the equationto compute the newly added sub-matrix. We also modify the proposed algorithm 1into the proposed algorithm 2, which is equivalent to the existing BLSalgorithm [7] in terms of numerical computations. The proposed algorithms 1 and2 can reduce the computational complexity, since usually the Hermitian matrixin the ridge inverse is smaller than the ridge inverse. With respect to theexisting BLS algorithm, the proposed algorithms 1 and 2 usually require about13 and 2 3 of complexities, respectively, while in numerical experiments theyachieve the speedups (in each additional training time) of 2.40 - 2.91 and 1.36- 1.60, respectively. Numerical experiments also show that the proposedalgorithm 1 and the standard ridge solution always bear the same testingaccuracy, and usually so do the proposed algorithm 2 and the existing BLSalgorithm. The existing BLS assumes the ridge parameter lamda->0, since it isbased on the generalized inverse with the ridge regression approximation. Whenthe assumption of lamda-> 0 is not satisfied, the standard ridge solutionobviously achieves a better testing accuracy than the existing BLS algorithm innumerical experiments.

Efficient Ridge Solution for the Incremental Broad Learning System on Added Nodes by Inverse Cholesky Factorization of a Partitioned Matrix